Large deviations of time-averaged statistics for Gaussian processes

In this paper we study the large deviations of time averaged mean square displacement (TAMSD) for Gaussian processes. The theory of large deviations is related to the exponential decay of probabilities of large fluctuations in random systems. From the mathematical point of view a given statistics satisfies the large deviation principle, if the probability that it belongs to a certain range decreases exponentially. The TAMSD is one of the main statistics used in the problem of anomalous diffusion detection. Applying the theory of generalized chi-squared distribution and sub-gamma random variables we prove the upper bound for large deviations of TAMSD for Gaussian processes. As a special case we consider fractional Brownian motion, one of the most popular models of anomalous diffusion. Moreover, we derive the upper bound for large deviations of the estimator for the anomalous diffusion exponent.